%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[12pt,t,aspectratio=169,mathserif]{beamer}
%Other possible values are: 1610, 149, 54, 43 and 32. By default, it is to 128mm by 96mm(4:3).
%run XeLaTeX to compile.

\input{wang-slides-preamble.tex}

\begin{document}

\title{8-{\ppr Black-Scholes} 模型 }
%\institute{上海立信会计金融学院}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年1月6日} }

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{1.1.1. }
\begin{frame}{内容提要 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容


\begin{enumerate}
\item[8.1.]  金融学
\item[8.6.]  什么是期权？
\item[8.8.]  期权定价问题的数学描述
\item[8.11.]  {\ppr Black-Scholes} 公式 
\end{enumerate}

%\item  一个有用的方法：测度变换
%\begin{enumerate}
%\item  什么是测度变换？
%\item  用测度变换来解释 {\ppr Black-Scholes} 公式
%\end{enumerate}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.1. 金融学 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  自从 {\ppr 1973} 年 {\ppr Black-Scholes} 和 {\ppr Merton} 的论文发表以来，随机分析在风险资产的定价中得到了广泛的应用。
\item  这里将解释金融的一些基本概念：债券、股票、期权、投资组合、波动率、交易策略、对冲、到期日、自融资、套利。
\item  在金融的研究文献里，等价鞅测度的概念和测度变换的方法也经常出现。
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.2. {\ppr A Short Excursion into Finance} }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  设风险资产的价格 $X_t$ 符合几何布朗运动模型
\begin{eqnarray*}
X_t = f(t,B_t) = X_0 \exp\left[ (c-\frac{1}{2}\sigma^2)t+\sigma B_t \right],
\end{eqnarray*}
其中 $\{B_t,t\ge 0\}$ 是标准布朗运动。

\item  几何布朗运动是下述随机微分方程的解
\begin{eqnarray*}
X_t = X_0 + c\int_0^t X_sds + \sigma \int_0^t X_sdB_s.
\end{eqnarray*}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.3.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  上述随机微分方程也经常写成
\begin{eqnarray*}
dX_t = cX_tdt + \sigma X_tdB_t.
\end{eqnarray*}

\item   在较短的时间区间 $[t,t+dt]$ 内，将上式写成差分形式，可得
\begin{eqnarray*}
X_{t+dt}-X_t &=& cX_tdt + \sigma X_tdB_t, \\
\frac{X_{t+dt}-X_t}{X_t} &=& cdt + \sigma dB_t. 
\end{eqnarray*}

\item  上式左边是相对回报率。
\item  $c$ 称为平均回报率。
\item  $\sigma$ 称为波动率。

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.4.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  无风险资产（债券）符合的微分方程、积分方程和价格公式分别为 
\begin{eqnarray*}
d\beta_t &=&  r \beta_tdt, \\ 
\beta_t &=& \beta_0 + r\int_0^t \beta_sds, \\ 
\beta_t &=& \beta_0 e^{rt}. 
\end{eqnarray*}

\item  设在 $t$ 时刻持有 $a_t$ 份股票和 $b_t$ 份债券，这是一个简单的投资组合。
\item  一个投资策略是指 $(a_t,b_t), t\in [0,T]$. 
\item  这个投资策略的财富过程是 
\begin{eqnarray*}
V_t=a_tX_t + b_t\beta_t,  t\in [0,T]. 
\end{eqnarray*}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.5.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  称投资策略是自融资的，如果财富的变化仅来自股票和债券价格的变化。用数学来定义自融资，是指成立
\begin{eqnarray*}
dV_t=a_tdX_t + b_td\beta_t,  t\in [0,T]. 
\end{eqnarray*}

\item  写成积分形式，在自融资策略下有下述成立
\begin{eqnarray*}
V_t - V_0 = \int_0^t a_sdX_s + \int_0^t b_Sd\beta_s. 
\end{eqnarray*}

\item  投资组合在 $t$ 时刻的财富等于初始投资 $V_0$ 加上股票和债券在 $[0,t]$ 时间内的收益。


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.6. 什么是期权？ }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  购买一份欧式看涨期权，在到期日 $T$ 带来的收益是
\begin{eqnarray*}
(X_T-K)^+ = \left\{
\begin{array}{ll}
X_T-K, & \text{ if }\,\, X_T>K, \\
0, & \text{ if }\,\, X_T\le K. 
\end{array}\right. 
\end{eqnarray*}

\item  购买一份欧式看跌期权，在到期日 $T$ 带来的收益是
\begin{eqnarray*}
(K-X_T)^+ = \left\{
\begin{array}{ll}
K-X_T, & \text{ if }\,\, X_T<K, \\
0, & \text{ if }\,\, X_T\ge K. 
\end{array}\right. 
\end{eqnarray*}

\item  期权定价问题：在 $t=0$ 时刻，这份期权的价格是多少？

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.7.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  {\ppr Black Scholes Merton} 的思路：
\begin{itemize}
\item  投资者在 $t=0$ 时刻投资购买这份期权，跟在 $t=0$ 时刻投资购买股票和债券，在到期日 $t=T$ 的收益是一样的。
\item  如果这份期权的价格过高或过低，将导致套利机会。
\end{itemize}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.8. {\ppr A Mathematics Formulation} }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  {\color{red}对冲未定权益 $(X_T-K)^+$ 是指寻找投资策略 $(a_t,b_t)$ 使得
\begin{eqnarray*}
\left\{\begin{array}{rcl}
V_t  &=& a_tX_t + b_t\beta_t = u(T-t, X_t), \,\, t\in [0,T], \\
V_T &=& u(0,X_T) = (X_T-K)^+.  
\end{array}\right.
\end{eqnarray*}
}

\item  记 $f(t,x)=u(T-t, x)$, 则 $V_t = f(t, X_t)$. 设 $X_t$ 满足随机微分方程
\begin{eqnarray*}
X_t &=& X_0 + c\int_0^t X_sds + \sigma \int_0^t X_sdB_s. \\
dX_t &=& cX_tdt + \sigma X_t dB_t.  
\end{eqnarray*}

\item  由 $\beta_t = \beta_0 e^{rt}$ 可得
\begin{eqnarray*}
d\beta_t = r\beta_t dt.  
\end{eqnarray*}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.9.  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  一方面，由伊藤公式，可得
\begin{eqnarray*}
V_t - V_0 &=& \int_0^t \left[ f_1(s,X_s) + cX_sf_2(s,X_s)+\frac{1}{2}\sigma^2 X_s^2 f_{22}(s,X_s)\right] ds \\ 
&& + \int_0^t \left[ \sigma X_sf_2(s,X_s) \right] dB_s. 
\end{eqnarray*}

\item  另一方面，由对冲的投资组合，可得
\begin{eqnarray*}
V_t - V_0 = \int_0^t a_sdX_s + \int_0^t b_s d\beta_s. 
\end{eqnarray*}

\item  上式右边第二项中的积分表达式可以写成
\begin{eqnarray*}
b_s d\beta_s = \frac{V_s-a_sX_s}{\beta_s} r\beta_sds = r(V_s-a_sX_s)ds. 
\end{eqnarray*}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.10. {\ppr The Black and Scholes Equation} }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  分别比较 $V_t-V_0$ 中的黎曼积分部分和伊藤积分部分，可得偏微分方程
\begin{eqnarray*}
u_1(t,x) = \frac{1}{2}\sigma^2x^2 u_{22}(t,x) + rxu_2(t,x) - ru(t,x),\,\,\, x>0, t\in [0,T]. 
\end{eqnarray*}

\item  边值条件为 
\begin{eqnarray*}
u(0,x) = (x-K)^+, x>0. 
\end{eqnarray*}

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.11. {\ppr The Black and Scholes Formula}  }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  {\ppr BS} 方程的解析解为
\begin{eqnarray*}
u(t,x) = x\Phi(g(t,x)) - Ke^{-rt}\Phi(h(t,x)). 
\end{eqnarray*}

\item  其中 $g(t,x), h(t,x)$ 和 $\Phi(x)$ 分别为
\begin{eqnarray*}
g(t,x) &=& \frac{\ln(x/K) + (r+ \frac{1}{2} \sigma^2)t}{\sigma\sqrt{t}}, \\ 
h(t,x) &=& \frac{\ln(x/K) + (r- \frac{1}{2} \sigma^2)t}{\sigma\sqrt{t}}, \\ 
\Phi(x) &=& \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp\left(-\frac{y^2}{2}\right)dy. 
\end{eqnarray*}


\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{8.12.   }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{itemize}%\itemsep1em

\item  {\color{red}欧式看涨期权在 $t=0$ 时刻的合理价格为 
\begin{eqnarray*}
V_0 = u(T,X_0) = X_0\Phi(g(T,X_0)) - Ke^{-rT} \Phi(h(T,X_0)). 
\end{eqnarray*}
}
\item  欧式看涨期权在 $t\in [0,T]$ 时刻的合理价格由下述随机过程给出，
\begin{eqnarray*}
V_t = u(T-t, X_t).
\end{eqnarray*}

\item  自融资投资策略 $(a_t,b_t)$ 由下述给出，
\begin{eqnarray*}
a_t = u_2(T-t, X_t), \,\,\,
b_t = \frac{u(T-t, X_t) - a_tX_t}{\beta_t}.
\end{eqnarray*}


\end{itemize}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}[fragile=singleslide]{1.20. }
\begin{frame}{参考文献}

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
%每页详细内容

\begin{thebibliography}{99}
%\bibitem{ssk} 司守奎, 孙玺菁. \emph{Python数学实验与建模}, 科学出版社. 2020年4月第1版.
%\bibitem{jiangqiyuan} 姜启源, 谢金星, 叶俊. \emph{数学模型}, 高等教育出版社. 2018年5月第5版.
%\bibitem{giordano} [美] Frank R. Giordano, William P. Fox, Steven B. Horton 著, 叶其孝, 姜启源等译. \emph{数学建模（原书第5版）} [A First Course in Mathematical Modeling (Fifth Edition)], 机械工业出版社, 2014年10月第1版. 
\bibitem{mikosch} {\ppr Thomas Mikosch}. {\ppr Elementary Stochastic Calculus}. 世界图书出版公司，{\ppr 2009} 年 {\ppr 8} 月第 {\ppr 1} 版。
\bibitem{wangjun} 王军、邵吉光、王娟. 随机过程及其在金融领域中的应用. 清华大学出版社，北京交通大学出版社，{\ppr 2018} 年{\ppr 8} 月第 {\ppr 2} 版。
\bibitem{zhangbo} 张波、商豪. 应用随机过程. 中国人民大学出版社，{\ppr 2016} 年 {\ppr 6} 月第 {\ppr 4} 版。
\bibitem{karlin} {\ppr Mark A. Pinsky, Samuel Karlin}. {\ppr An Introduction to Stochastic Modeling}. 机械工业出版社，{\ppr 2013} 年 {\ppr 2} 月第 {\ppr 1} 版。

\end{thebibliography}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


